A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations
Reads0
Chats0
TLDR
In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.Abstract:
In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.read more
Citations
More filters
Journal ArticleDOI
On a Classical Spin Glass Model
TL;DR: In this article, a simple exactly soluble model of a spin-glass with weakly correlated disorder is presented, which includes both randomness and frustration, but its solution can be obtained without replicas.
Journal ArticleDOI
Optimal Alarm Signal Processing: Filter Design and Performance Analysis
TL;DR: For the first problem, design of optimal linear FIR alarm filters is studied, and a numerical optimization based procedure is proposed.
Journal ArticleDOI
Average-case analysis of algorithms for matchings and related problems
TL;DR: This work shows that in almost every graph, any nonmaximum O–1 flow admits a short augmenting path, and proves that augmenting-path algorithms, that are fast in the worst case, also perform exceedingly well on the average.
Journal Article
A bound on the deviation probability for sums of non-negative random variables.
TL;DR: In this article, a simple bound is presented for the probability that the sum of nonnegative independent random variables is exceeded by its expectation by more than a positive number t. The inequality extends to one-sidedly bounded martingale difference sequences.
Journal ArticleDOI
Toward a theory of information processing
Sinan Sinanovic,Don H. Johnson +1 more
TL;DR: This work uses information-theoretic distance measures, the Kullback-Leibler distance in particular, to quantify how well signals represent information and derives the fundamental processing capabilities of simple system architectures that apply universally.
References
More filters